Digging Deeper into … Capacity (all classes)

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Digging Deeper into … Capacity (all classes)

Strictly speaking, capacity is the amount (or measure) of a substance (which can be solid, liquid or gas) that something can hold (i.e. a container). That said, in primary mathematics we tend to use capacity as a measure of liquids only (ie not solids or gases), both to avoid confusion and since the children would most commonly see examples of liquids measured using the standard units of capacity (ie litres and millilitres).

Initial exploration – CPA approach

Like the topics of Length and Weight, and in keeping with the over-arching CPA approach of Operation Maths, children’s initial experiences of capacity at every class level should focus on hands-on activities, using appropriate concrete materials.

In the younger classes, this should occur through exploration, discussion, and use of appropriate vocabulary eg full, nearly full, empty, holds more, holds less, holds as much as/the same as etc. The children should also be enabled to sort, compare and order containers according to capacity.

From Operation Maths 1

Irrespective of the class level, introductory exploration in this topic could follow the following progression or similar:

The children examine pairs of empty containers and make comparisons, so as to identify, from sight, which holds more/less. Use questioning to encourage them to assess all available information:

Which container is wider/narrower?

Which container is taller/shorter?

Elicit from the children how they might verify their estimates. Introduce a non-standard measure (e.g. egg-cup, yogurt container, plastic cap from an aerosol, tea/table spoons, plastic syringe, flask etc) and demonstrate how to measure the capacity of a container using a non-standard measure eg (using egg-cup as standard measure):

Fill an egg-cup with water. Pour this into the target container to be measured. Repeat until container is full and then record the number of egg-cups required.

OR fill the container with water. From this, pour out an egg-cup full, which is then poured out into a third container (eg basin, plastic box). Repeat until the target container is empty and then record the number of egg-cups that were filled from it.

OR fill the target container with water. Pour this into a larger container and record the level of the water by marking the level on the side. Pour out the water out into a third container (eg basin, plastic box) to be used as a water store/reservoir. Repeat with other containers to be measured and use the marking on the side of the measuring container to identify which container held the most/least etc. Please note though, that while this method can be used to identify which container holds the most/least, it will not provide a measure of the capacity as a quantity of non-standard units (unless of course the measuring container has existing markings for litres and/or millilitres)

From Operation Maths 1

HINT: In order to be avoid unnecessary water wastage and/or a very wet classroom (!), it can be a good idea to conduct the capacity activities outside and over a number of plastic basins/boxes. These can be used to catch spills and to hold the water which can be re-used repeatedly to measure the capacity of the various containers. 20 ml or 50 ml plastic syringes can also be very useful; they are easy for smaller hands to use draw up water and squirt it into a container. And instead of counting ml, ask the children just to record the capacity of the container as the number (count) of syringes that it can hold.

Move on to pairs of containers whose difference in capacity may not be obvious because of the shape and dimension of the containers. Thus, it is important to use a selection of containers that vary in height and width.

This can then progress to incorporate a direct comparison of the capacity of three or more containers. It is important at this stage that the children realise that if A holds more than B and B holds more than C, then, without further direct comparisons, we know that A holds more than C, that A holds the most of all three and C holds the least. This is a very important concept for the children to grasp.

HINT: Use brainstorming to elicit the names of various liquids and container types with which the children are familiar. Use the list to make up an odd one out game, as outlined below

In a similar way, the children can estimate and record the capacity of containers of objects using standard units (i.e. litres and millilitres; the latter is introduced in third class). Initially, when using the standard unit of a litre (starting from first class) the children will be recording the capacity of containers as being able to hold more than/less than/the same as a litre.

HINT: In 2nd class & 3rd class the children will be using 1/2 litre and 1/4 litre (as opposed to millilitres). This will necessitate using bottles etc that are marked in 1/4 litre intervals. Challenge the children in these classes up to come up with ways to measure and mark these intervals, without having to use millilitres or some type of commercial graduated measure (eg a jug). This task could be given as an alternative homework activity.

When finding the capacity of a container, it is important also to highlight to the children that it is not necessary to fill it to the brim. Show them an example of an unopened litre bottle of water – the height of the water in the
bottle is not to the brim, yet the label shows it contains 1 litre. Thus, the children will develop an understanding that the actual capacity of containers are typically greater than the indicated capacity of the liquid it contains.

Problem Solving: How many are needed to fill? It takes 4 of container A to fill container B. It takes 2 of container B to fill container C. How many of container A are needed to fill C? This can be a very difficult concept to grasp for many children. Some suggestions include using multiples of the real containers to show the relationships between each and drawing pictorial representations using bar models, one of the three key visual strategies for problem-solving used throughout Operation Maths, (shown below).

Using more accurate measures

As the children progress in their understanding of the concept of capacity they will begin to appreciate the need for more accurate means to record it; both using smaller standard units (ie millilites) and using measures/containers which are already calibrated/graduated with markings. It is an advantage to have a wide selection of different types of measuring instruments available (including plastic jugs, syringes, measuring spoons, graduated cylinders etc) so that the children appreciate that different measuring instruments are more suitable for certain tasks. When measuring, advise the children also to read the level of liquid at eye level to obtain a more accurate reading.

HINT: Some jugs etc can be purchased relatively cheaply from value shops. Alternatively, ask the children to bring in measuring jugs, containers etc., from home to use in class while working on this topic.

As always, the children should be encouraged to estimate before measuring. And, rather than estimating the capacity of A, B, C and D before measuring A, B, C and D, it would be better if the children estimated the capacity of A and then measured the capacity of A, estimated the capacity of B and then measured the capacity of B and so on. Thus, they can reflect on the reasonableness of their original estimate each time and use this to refine their next estimate so that it might be more accurate. This helps them internalise a sense of capacity, and to use this sense to produce more accurate estimates.

When the children have experienced using a variety of instruments for measuring capacity, they should then be afforded the opportunity to choose which instrument (and which standard unit) is most appropriate to measure the capacity of various containers. In this way, the children start developing the notion that while many approaches can be taken, some are more efficient than others, and the most efficient approach will also depend on the target object being measured. This is the same as the Operation Maths approach to operations throughout the classes; there can be many approaches and some are more efficient than others, depending on the numbers/operations involved. The aim is for the children to become accurate, efficient and flexible thinkers.

Renaming units of capacity

From fourth class on, the children will be expected to rename units of capacity, appropriate to their class level. While changing 1,250 ml to 1 l 250ml or 1.25 l, will typically be done correctly, converting figures which require zero as a placeholder (eg 1 l 50 ml, 2.6 l ) can be more problematic, and can reveal an underlying gap in understanding, that is not revealed by the more obvious measures. In these cases, the children should be encouraged to return to the concrete experiences as a way of checking the reasonableness of their answers, eg:

“2.6 l equals 2,600 ml because 1 l is 1,000 ml, so 2 l is 2,000 ml and .6 is slightly more than .5, which is half of a l or 500 ml, which means .6 must be 600 ml”

T-charts, another one of the three key visual strategies for problem-solving used throughout Operation Maths, can be very useful when renaming units of capacity, as can be seen below. These can be partially started on a class board and the children then asked to complete the T-chart with their own choice of capacities as is relevant to the tasks required of them. The children could construct these also to use as a reference, as they progress through this topic.

Capacity & Volume

Volume is introduced officially for the first time in 6th class. It is preferable to introduce the children to volume via cubed units (eg blocks) as opposed to via cubed centimetres (see below).

From Operation Maths 6

HINT: Did you know that the smallest base-ten blocks (ie those often used as units or thousandths), are 1 cm cubed? This means that these could be used to build shapes from which the volume of the shapes can be measured and they can be used to measure the approximate volume of an open cuboid eg lunch box, pencil box, etc.

The children may find it challenging to appreciate the relationship between capacity and volume, especially since they may think capacity is exclusive to liquids while volume relates to solids. Providing the children with opportunities to measure the the capacity of a variety of different sized cuboids (eg lunch box) and then measuring its volume using 1 cm cubes, will likely lead the children to discover the connection between the two concepts and that 1cm cubed equals 1 ml.

From Operation Maths 6

Further Reading & Viewing:

Children in senior classes might be interested in researching the history of the metric system. This website could be a good place to start.

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